Abstract

Recently the insurance industry has started to realise the importance of properly managing options and guarantees embedded in insurance contracts. Interest rates have been low in the last few years, which means that minimum interest rate guarantees have moved from being far out-of-the money to expiring inthe-money. As a result, many insurance companies have experienced solvency problems. Furthermore, insurance rms operating within the European Union will, from the end of 2012, be subject to the Solvency II directive, which places new demands on insurance companies. For example, the valuation of assets and liabilities now needs to be market consistent. One way to accomplish a market consistent valuation is through the use of an economic scenario generator (ESG), which creates stochastic scenarios of future asset returns. In this thesis, we construct an ESG that can be used for a market consistent valuation of guarantees on insurance contracts. Bonds, stocks and real estate are modelled, since a typical insurance company's portfolio consists of these three assets. The ESG is calibrated to option prices, wherever these are available. An Otherwise the calibration is based on an analysis of historical volatility.

assessment of how well the models capture prices of instruments traded on the market is made, and nally the ESG is used to compute the value of a simple insurance contract with a minimum interest rate guarantee.

Acknowledgements

We would like to thank Handelsbanken Liv for making this thesis possible. Special thanks to Tobias Lindhe, our supervisor at Handelsbanken Liv, and our supervisor Boualem Djehiche at the Division of Mathematical Statistics at KTH for their guidance and support. We would also like to thank Fredrik Bohlin, Maja Ernemo, Rikard Kindell, Björn Laurenzatto and Jonas Nilsson for their helpful feedback and interesting discussions.

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Contents

1 Introduction

1.1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valuation of Liabilities under Solvercy II . . . . . . . . . . . . . . 1.2.1 1.2.2 1.3 QIS5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Best Estimate . . . . . . . . . . . . . . . . . . . . . . . .

1

1 2 2 3 4

Aim and Scope

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2

Theoretical Background

2.1 2.2 2.3 2.4 Economic Scenario Generators . . . . . . . . . . . . . . . . . . .

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5 6 7 8 10 12 13 14 15 15

Calibration of a Market Consistent ESG . . . . . . . . . . . . . . Overview of Models and Methods . . . . . . . . . . . . . . . . . . Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The Hull-White Extended Vasicek Model 2.4.1.1 2.4.2 . . . . . . . . .

Bond and Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Black-Karasinski Model

2.5

Equity Models 2.5.1 2.5.2

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Volatility in the Black-Scholes Model . . . . . . . . . . . . Interest Rates in the Black-Scholes Model . . . . . . . . . 2.5.2.1 2.5.2.2

The Equity Model under Hull-White Interest Rates 16 The Equity Model under Black-Karasinski Interest Rates . . . . . . . . . . . . . . . . . . . . 16 17 18

2.6

Real Estate Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.6.2 2.7

Unsmoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18 20 20 21

Correlation and Simulation 2.7.1 2.7.2 Correlation

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Monte Carlo Simulation . . . . . . . . . . . . . . . . . . .

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Analysis

3.1 Interest Rate Modelling 3.1.1 . . . . . . . . . . . . . . . . . . . . . . .

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22 22 22 24 27 30 33 33 33 34 36 39 39 41...